How the garden bridge will look, according to its backers. Photograph: Arup/Heatherwick Studios

You get a cold call from someone charming and plausible, who tells you about something wonderful – solar panels, let’s say, or a conservatory – that will transform your house, give a faint glow of green virtue and which due to some miraculous financing arrangement will cost you nothing. Feeling weak, you say yes, and end up with a leaking roof, unmet promises and large bills. Something like this is happening to the public with the garden bridge proposed for the Thames.

Lord Davies, chairman of the Garden Bridge Trust, says that “support for the bridge has been overwhelming”. You would expect this: who could not want the free gift of a poetic-vision-cum-infrastructural-facility? Except that such public consultation as has happened is rendered worthless by the fact that it was vague or silent on a number of significant details, which might well have led to different responses. The small print for the Amazing Garden Bridge Offer is itself fantastical.

Presented last year as a £60m project entirely funded by the private sector, it is now has a budget of £175m and requires £60m of public funding, money that could otherwise be spent on much needed crossings of less privileged parts of the Thames. Described by its inventor, Joanna Lumley, as “a chance to walk through woodlands”, it actually has 2,500 square metres of vegetation, which is less than half a football pitch, and is reached by hefty stairs and ramps. In the space of three months, its estimated annual running costs have gone from £2.5m to £3m to £3.5m.

It is called a place to “linger and enjoy the river and all of its attributes in a peaceful and tranquil environment”, but Transport for London has estimated that the football match-size crowds around its landing point will sometimes have a “pedestrian comfort level” of D – this is the second worst of five categories and means “very uncomfortable”. The project will create potentially dangerous crowd pressures on nearby parts of the southern Thames embankments that haven’t been studied. On the bridge itself, numbers will be controlled by security, with resulting queues to get on. This is hardly a ramble of Wordsworthian bliss.

It has also been called “a new public park space”, but last week it emerged that groups larger than eight will have to notify their intention to enter the bridge in advance, that there will be no public right of way across it, that it will close at midnight and that it will be taken over 12 times a year by money-earning events. Picnics are prohibited, as is anything resembling a political protest. Cycling is impossible and facilities for bike parking are limited. No additional lavatories are planned for the expected 7.1 million visitors a year. It requires the destruction of 30 mature trees and of existing green space. “Thousands of local residents,” according to objectors, “will have their lives blighted.”

The discoveries continue. Those £3.5m running costs are to be paid for in part by “pop-up type events” and “a discreet range of merchandise”. In other words, the bridge’s approaches will be infested with advertising and selling. There is an ambitious programme of fundraising from patrons and corporate members, who will surely require that their generosity is recognised by prominent display of their names. £550,000 is supposed to be raised from a gala dinner every year for the entire existence of the bridge. It is barely credible that, year after year into an indefinite future, benefactors will stump up these sums for annual maintenance costs. If they don’t, the public purse will be asked to pay. According to the lawyers of the Middle Temple, whose buildings will be affected, there is “unacceptable uncertainty of income stream”.

Earlier this month, the borough of Lambeth’s planning committee narrowly granted planning permission, subject to 46 conditions that reflect the project’s swarm of vagueness. The council breezily said that “management and maintenance” will be “key to success” but approved the scheme without conclusive evidence that these would or could be satisfactorily resolved. The bridge also requires permission from the City of Westminster, whose planning committee will decide on 2 December. Its leader, Philippa Roe, has already made her enthusiasm for the bridge clear.

By any reasonable interpretation of the relevant planning policies, the bridge should be refused. It disrupts protected views of St Paul’s and has a huge impact on the open character of the Thames. It also affects the setting of Somerset House and other heritage assets. The structures where it lands are crude impositions on a conservation area. The Middle Temple believes that it is unlawful to grant permission when there are so many uncertainties about crucial issues. The main justification for overruling these objections is what Lambeth called the bridge’s “iconic appearance and captivating silhouette” and the idea that it will be a delightful place to linger. To believe it will be so means imagining that tranquillity can be achieved amid crowds you might find at a football ground.

It means overlooking the significant number of people who find the structure ugly and egotistical. It also means hoping that all promises on funding and management will be fulfilled. It requires belief that the trees and vegetation will, in an exposed and challenging situation, be as lush and abundant as they are in the enticing visualisations. It also means putting complete faith in the Garden Bridge Trust, the bridge’s designer, Thomas Heatherwick, and its enthusiastic backer Boris Johnson. Yet the trust’s half-truths and the project’s record of unpleasant surprises don’t inspire confidence. The bridge resembles other Johnson-backed projects such as the Orbit sculpture in the Olympic Park and the sponsored cable car across the Thames that haven’t turned out as wonderfully as promised.

Heatherwick is an inventive and talented product designer whose public projects sometimes suffer glitches. His B of the Bang sculpture in Manchester was dismantled after it started shedding metal, and his Blue Carpet in Newcastle was late and over budget and in the space of a few years became grey and shabby. His London buses overheat. This record raises reasonable doubts whether his bridge will be everything now promised.

The bridge’s fans back it with good intentions. They want something beautiful. They believe that a great city shouldn’t stand still. They are right to explore ideas like this. But there comes a time to recognise that the thing they love is not what they thought it would be, that their nice idea just doesn’t work. It is not a tranquil walk in woodlands. It is not a genuinely public place. It is not free. It is not a well-conceived piece of transport infrastructure. It is a crowded and overstyled chunk of heavy engineering garnished with urban parsley.